Aim: What are Riemann Sums?

1. Aim: What are Riemann Sums? Do Now: Approximate the area under the curve y = 4 – x2 for [-1, 1] using 4 inscribed rectangles.

2. Devising a Formula • Using left endpoint to approximate area under the curve is the more rectangles the better the approximation lower sum yn - 1 the exact area? take it to the limit! y1 yn - 1 y0 yn - 2 2 1 a b left endpoint formula

3. Right Endpoint Formula • Using right endpoint to approximate area under the curve is yn upper sum yn - 1 right endpoint formula y1 y0 midpoint formula a b

4.  where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation. Sigma Notation sum of terms sigma The sum of the first n terms of a sequence is represented by

5. Summation Formulas

6. Δx4 Δx5 Δx6 Δx1 Δx2 Δx3 x0 x1 x2 x3 x4 x5 x6 Riemann Sums • A function f is defined on a closed interval [a, b]. • It may have both positive and negative values on the interval. • Does not need to be continuous. Partition the interval into n subintervals not necessarily of equal length. a = x0 < x1 < x2 < . . . < xn – 1< xn = b a = = b Δxi = xi – xi – 1 - arbitrary/sample points for ith interval

7. Δx4 Δx5 Δx6 Δx1 Δx2 Δx3 Riemann Sums • Partition interval into n subintervals not necessarily of equal length. x0 a = x1 x2 x3 x4 x5 x6 = b - arbitrary/sample points for ith interval ci = xi

8. Δx6 Δx4 Δx2 Δx1 Riemann Sums Δxi = xi – xi – 1 x6 x0 a = = b

9. Definition of Riemann Sum Let f be defined on the closed interval [a, b], and let Δ be a partition of [a, b] given by a = x0 < x1 < x2 < . . . . < xn – 1 < xn = b, where Δxi is the length of the ith subinterval. If ci is any point in the ith subinterval, then the sum is called a Riemann sum for f for the partition Δ largest subinterval – norm - ||Δ|| or |P| equal subintervals – partition is regular regular partition general partition converse not true

10. Model Problem Evaluate the Riemann Sum RP for f(x) = (x + 1)(x – 2)(x – 4) = x3 – 5x2 + 2x + 8 on the interval [0, 5] using the Partition P with partition points 0 < 1.1 < 2 < 3.2 < 4 < 5 and corresponding sample points

11. Model Problem

12. Definition of Definite Integral If f is defined on the closed interval [a, b] and the limit exists, the f is integrable on [a, b] and the limit is denoted by The limit is called the definite integral of f from a to b. The number a is the lower limit of integration, and the number b is the upper limit of integration. Definite integral is a number Indefinite integral is a family of functions If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b].

13. Evaluating a Definite Integral as a Limit ci = xi

14. Evaluating a Definite Integral as a Limit not the area The Definite Integral as Area of Region If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded by the graph of f, the x-axis and the vertical lines x = a and x = b is given by

15. Properties of Definite Integrals

16. Areas of Common Geometric Figures Sketch & evaluate area region using geo. formulas. = 8 A = lw

17. Model Problems =0

18. Model Problem

19. A2 A1 Total Area = -A1 + A2 Model Problem

20. A2 A1 Model Problem

21. Model Problem take the limit n

22. Definition of Riemann Sum

23. Definition of Riemann Sum

24. Subintervals of Unequal Lengths

25. Subintervals of Unequal Lengths